Book contents
- Frontmatter
- Contents
- Introduction
- XV Tubular extensions and tubular coextensions of algebras
- XVI Branch algebras
- XVII Tilted algebras of Euclidean type
- XVIII Wild hereditary algebras and tilted algebras of wild type
- XIX Tame and wild representation type of algebras
- XX Perspectives
- Bibliography
- Index
- List of symbols
XVII - Tilted algebras of Euclidean type
Published online by Cambridge University Press: 12 January 2010
- Frontmatter
- Contents
- Introduction
- XV Tubular extensions and tubular coextensions of algebras
- XVI Branch algebras
- XVII Tilted algebras of Euclidean type
- XVIII Wild hereditary algebras and tilted algebras of wild type
- XIX Tame and wild representation type of algebras
- XX Perspectives
- Bibliography
- Index
- List of symbols
Summary
We have already described in Chapter XII the structure and the combinatorial invariants of the module categories mod A of concealed algebras A of Euclidean type, while in Chapter XIV we give a complete classification of these algebras A by means of quivers and relations.
The main objective of this chapter is to describe the structure and the combinatorial invariants of the module category mod B of an arbitrary representation-infinite tilted algebra B of Euclidean type. Moreover, we show that these algebras B are domestic branch extensions or domestic branch coextensions of concealed algebras A of Euclidean type. In Section 1, we study the distribution of indecomposable direct summands of a splitting tilting module among the hereditary standard stable tubes of the Auslander–Reiten quiver Γ(mod B) of an arbitrary algebra B, while in Section 2 we show how the structure of the hereditary standard stable tubes in Γ(mod B) is changed under the related tilting process of B.
The main result of Section 3 asserts that every representation-infinite tilted algebra B of Euclidean type is a domestic tubular (branch) extension or a domestic tubular (branch) coextension of a concealed algebra A of Euclidean type. The inverse implication is proved in Section 4 by showing that every domestic tubular (branch) extension and every domestic tubular (branch) coextension of a concealed algebra of Euclidean type is a representation-infinite tilted algebra of Euclidean type.
In Section 5, we present a characterisation of representation-infinite tilted algebras B of Euclidean type, and we exhibit their module categories mod B.
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- Elements of the Representation Theory of Associative Algebras , pp. 103 - 212Publisher: Cambridge University PressPrint publication year: 2007